Question

21.(10 pts.) Suppose f(x) is a function that has its only singularities at the integers. In other words, Zx is a singularity for f(z) if and only if zx = m. m e Z. Further suppose that the residue off at every even integer is -4 and that the residue off at every odd integer is 1. Sketch a contour on which the integral off would be equal to -22ri. Explain your reasoning.

Answer 1

et such that c be a contour SF2142 = azzal I Ha= 22* residue theorem, 2d3 z zri R q residues of fiz) where R is the sum at pals inside c. Thu, axir = -22 ti ilieSum que observe residues that =-11 all = 36-4)+1(1) = 3x Residue at even poles tix Residus at odd poles, Thu, c is a contour that contains two B even integers and I odd integer inside a it. There are infinitely many contours with this property, one such contour is shown in figure w Res (shains . even poles Chamely 2014 2-2 and 3=4) and / one odd bole chamely 1 of 2 2x 2=1).